Exercise 3
Yuting Huang, Jipeng Cheng, Weidi Hou
4/5/2022
Problem 1: What causes what?
Question 1. Why can’t I just get data from a few different cities and run the regression of “Crime” on “Police” to understand how more cops in the streets affect crime? (“Crime” refers to some measure of crime rate and “Police” measures the number of cops in a city.)
answer: In this question of “Crime” and “Police”, there is a problem of endogeneity circle. if we use the simple linear regresstion model \[y=beta_1+beta_2 * x+u\], there is a problem of endogeneity, i.e. E(xu)≠0. Under is circumstance, inconsistency of OLS arises due to the fact that changes in x(police force) are associated not only with chages in y(crime), but also change in u.
Question2. How were the researchers from UPenn able to isolate this effect? Briefly describe their approach and discuss their result in the “Table 2” below, from the researchers’ paper.
answer: In the paper, the authors use a “high-alert periods” dummy variable to break the circle of endogeneity to estimate the effect of police on crime. The reason why the authors choose “high-alert periods” is that the primary purpose of the HSAS is to inform and coordinate the antiterrorism efforts of all federal agencies. So, it the level of alert will directly impact the number of police on the specific district. In addition, the authors use daily data to make sure the “treatment windows” are short. Plus, the choosing data includes the information repeated terror alert, so it reduces the possibility of spurious correlation. Furthermore, the model also uses a “Metro ridership” variable to test whether there is a correlation between tourism and crime. At last, the authors use dummy variables for each day of the week to control for day effects.
From the Table 2 column 1, the coefficient on the alert level is statistically significant at the 5 percent level and indicates that on high-alert days, total crimes decrease by an average of seven crimes per day, or approximately 6.6 percent. From the Table 2 column 2, we verify that high-alert levels are not being confounded with tourism levels by including logged midday Metro ridership directly in the regression. The coefficient on the alert level is slightly smaller, at -6.2 crimes per day. We also find that increased Metro ridership is correlated with an increase in crime. The increase, however, is very small, a 10 percent increase in Metro ridership increases the number of crimes by only 1.7 per day on average. Thus, given that midday Metro ridership is a good proxy for tourism, changes in the number of tourists cannot explain the systematic change in crime that we estimate.
Question3. Why did they have to control for Metro ridership? What was that trying to capture?
In order to test a hypothesis that tourism is reduced on high-alert days, and as a result, there are fewer potential victims, which leads to fewer crimes.
Question4. Below I am showing you “Table 4” from the researchers’ paper. Just focus on the first column of the table. Can you describe the model being estimated here? What is the conclusion?
The model in table 4 includes district fixed effects in order to distinguish the peculiar crime pattern of each district, and all regressions contain day-of-the-week fixed effects. The dependent variable is daily crime totals by district. From Table 4, During periods of high alert, crime in the National Mall area decreases by 2.62 crimes per day. Crime also decreases in the other districts, by 0.571 crimes per day, but this effect is not statistically significant. Since there are 17.1 crimes on the district 1, the declination during high-alert days is approximately 15 percent, which means almost one-half of the total crime decline during high-alert periods is concentrated in District 1. In addition, The result elasticity of crime with respect to police is -0.3, which is consistent with other researchers’ results.
Problem 2: Tree modeling: dengue cases
## Distribution not specified, assuming gaussian ...| dengue_tree | dengue_forest | dengue_boost |
|---|---|---|
| 39.00296 | 39.13611 | 38.83203 |
Use Kable to name columns in .Rmd. The results suggest that random forests mode have the best performance on the testing data.
Problem 3: Predictive model building: green certification
Overview
As landlords and renters, what they mostly care about is their possible revenue per square foot per calendar year. As we all know, their revenue depends on many factors, such as the age, facilities, amenities , leasing rate and so on. Nowadays, people pay more attention on their living environment when renting a house, such as whether this landload has a green certification. Therefore, it’s meaningful to do some research on the relationship between rental income and green certification.
In our report, we build our best predictive model possible for revenue per square foot per calendar year and to use this model to quantify the average change in rental income per square foot associated with green certification, holding other features of the building constant.
Data and research design
Data
In our raw data set, there are 7,894 data points. We first filter out those missing data. Now, our new greenbuildings data set has 7820 data points.
predictive variable and features
The predictive variable is revenue per square foot per year, which is the product of two terms: rent and leasing_rate.
The features used to build our model are:
(1) CS.PropertyID: the building's unique identifier in the database.
(2) cluster: an identifier for the building cluster, with each cluster containing one green-certified building and at least one other non-green-certified building within a quarter-mile radius of the cluster center.
(3) size: the total square footage of available rental space in the building.
(4) empl.gr: the year-on-year growth rate in employment in the building's geographic region.
(5) Rent: the rent charged to tenants in the building, in dollars per square foot per calendar year.
(6) leasing.rate: a measure of occupancy; the fraction of the building's available space currently under lease.
(7) stories: the height of the building in stories.
(8) age: the age of the building in years.
(9) renovated: whether the building has undergone substantial renovations during its lifetime.
(10) (11) class.a, class.b: indicators for two classes of building quality (the third is Class C). These are relative classifications within a specific market. Class A buildings are generally the highest-quality properties in a given market. Class B buildings are a notch down, but still of reasonable quality. Class C buildings are the least desirable properties in a given market.
(12) green.rating: an indicator for whether the building is either LEED- or EnergyStar-certified.
(13) LEED, Energystar: indicators for the two specific kinds of green certifications.
(14) net: an indicator as to whether the rent is quoted on a "net contract" basis. Tenants with net-rental contracts pay their own utility costs, which are otherwise included in the quoted rental price.
(15) amenities: an indicator of whether at least one of the following amenities is available on-site: bank, convenience store, dry cleaner, restaurant, retail shops, fitness center.
(16) cd.total.07: number of cooling degree days in the building's region in 2007. A degree day is a measure of demand for energy; higher values mean greater demand. Cooling degree days are measured relative to a baseline outdoor temperature, below which a building needs no cooling.
(17) hd.total07: number of heating degree days in the building's region in 2007. Heating degree days are also measured relative to a baseline outdoor temperature, above which a building needs no heating.
(18) total.dd.07: the total number of degree days (either heating or cooling) in the building's region in 2007.
(19) Precipitation: annual precipitation in inches in the building's geographic region.
(20) Gas.Costs: a measure of how much natural gas costs in the building's geographic region.
(21)Electricity.Costs: a measure of how much electricity costs in the building's geographic region.
(22)City_Market_Rent: a measure of average rent per square-foot per calendar year in the building's local market.research design
## [1] 7.426478## Distribution not specified, assuming gaussian ...## [1] 9.387564By using the tree model to build our predictive model, we identify the association between rental revenue and green certification. We first split the new greenbuilds data set into training and test test, then we use two different tree methods: random forest and boosted regression trees to build our model. Then we calculate the out of sample RMSE. The result shows that the RMSE of the random forest model is lower that the one using boosted regression trees. Therefore, we choose to use random forest method.
Results
Figure 1: Figure of all features used in our predictive model descending by importance.
From Figure 1, we found that the most important features that landlords should consider are “size, age, stories”, these variables seems to be the most important. However, the “green_rating” variable is not that important from our observation.
Figure 2: Impact of Green Certifications on Revenue
From the box plot including the information between revenue and green_rating, the average rent per square foot is around 20 for buildings with a green certification, and around 24 for those without a certification. Therefore, we didn’t find significant difference between buildings with or without green certification.
## # A tibble: 2 × 2
## green_rating `mean(predict_rent)`
## <int> <dbl>
## 1 0 23.9
## 2 1 26.6Table 1: The average change in revenue per square foot related to green certification
We then calculate the average change in revenue per square foot for buildings with or without green certification. From the table above, we found that buildings with green certification get more revenue than those without green certification, but the impact is small, considering the difference is 2 unit.
Figure 3: Partial Effect of Green Certifications on Revenue
From the partial effect figure of green certification on revenue per square foot, we found a positive association between revenue and green certification, which means green certification has some good impact on revenue.
Conclusion
From our analysis, we build the model using random forest to estimate the association between green certification and revenue per square foot. From our results, we found that green certification has some positive impact on revenue, but the impact isn’t very significant. In addition, we found that these variables “size, age, stories” have more impact on revenue. Based on our research, the landlords should weigh the pros and cons of getting a green certification, thinking critically about the cost and revenue change of getting a green certification. If they figure that the income change isn’t overweigh the cost of getting a certification, they should focus on other features, which have more important impact on revenue.
Problem 4: Predictive model building: California housing
Our purpose is to build the best predictive model to forecast the median house value in California. According to the topic description, in the beginning, we have to conduct standardized processing for “totalRooms” and totalBedrooms" by creating the new variables including “sdrooms” and “sdbedrooms”. Then we utilized three different methods including Step wise, Random Forecast and boost, to build the best predictive model.
## longitude latitude housingMedianAge totalRooms totalBedrooms population
## 1 -122.23 37.88 41 880 129 322
## 2 -122.22 37.86 21 7099 1106 2401
## 3 -122.24 37.85 52 1467 190 496
## 4 -122.25 37.85 52 1274 235 558
## 5 -122.25 37.85 52 1627 280 565
## 6 -122.25 37.85 52 919 213 413
## households medianIncome medianHouseValue
## 1 126 8.3252 452600
## 2 1138 8.3014 358500
## 3 177 7.2574 352100
## 4 219 5.6431 341300
## 5 259 3.8462 342200
## 6 193 4.0368 269700##
## Call:
## lm(formula = medianHouseValue ~ longitude + latitude + housingMedianAge +
## sdrooms + sdbedrooms + population + medianIncome, data = cahousing_train)
##
## Coefficients:
## (Intercept) longitude latitude housingMedianAge
## -3.663e+06 -4.309e+04 -4.163e+04 9.214e+02
## sdrooms sdbedrooms population medianIncome
## -1.091e+04 6.031e+04 -1.120e+00 4.366e+04## Start: AIC=323327
## medianHouseValue ~ longitude + latitude + housingMedianAge +
## sdrooms + sdbedrooms + population + medianIncome
##
## Df Sum of Sq RSS AIC
## + sdbedrooms:population 1 8.1291e+11 7.4853e+13 323173
## + sdrooms:sdbedrooms 1 6.5835e+11 7.5008e+13 323203
## + housingMedianAge:sdbedrooms 1 2.8920e+11 7.5377e+13 323274
## + housingMedianAge:sdrooms 1 2.7907e+11 7.5387e+13 323276
## + housingMedianAge:medianIncome 1 2.7099e+11 7.5395e+13 323277
## + population:medianIncome 1 1.6003e+11 7.5506e+13 323298
## + latitude:sdbedrooms 1 1.3840e+11 7.5528e+13 323303
## + sdrooms:medianIncome 1 1.3125e+11 7.5535e+13 323304
## + longitude:sdbedrooms 1 9.3743e+10 7.5573e+13 323311
## + latitude:medianIncome 1 9.0990e+10 7.5575e+13 323312
## + longitude:housingMedianAge 1 7.6050e+10 7.5590e+13 323315
## + longitude:sdrooms 1 3.6121e+10 7.5630e+13 323322
## + latitude:housingMedianAge 1 3.4087e+10 7.5632e+13 323323
## + sdbedrooms:medianIncome 1 3.2417e+10 7.5634e+13 323323
## + sdrooms:population 1 2.3319e+10 7.5643e+13 323325
## + longitude:population 1 1.3807e+10 7.5653e+13 323326
## <none> 7.5666e+13 323327
## + latitude:population 1 9.4775e+09 7.5657e+13 323327
## + longitude:latitude 1 5.4158e+09 7.5661e+13 323328
## + housingMedianAge:population 1 2.5299e+09 7.5664e+13 323329
## + longitude:medianIncome 1 1.4852e+09 7.5665e+13 323329
## + latitude:sdrooms 1 1.2559e+08 7.5666e+13 323329
## - population 1 2.0994e+10 7.5687e+13 323329
## - sdrooms 1 1.2061e+12 7.6872e+13 323553
## - housingMedianAge 1 1.5667e+12 7.7233e+13 323621
## - sdbedrooms 1 1.7947e+12 7.7461e+13 323664
## - longitude 1 1.1817e+13 8.7483e+13 325421
## - latitude 1 1.2071e+13 8.7737e+13 325463
## - medianIncome 1 3.8896e+13 1.1456e+14 329317
##
## Step: AIC=323173
## medianHouseValue ~ longitude + latitude + housingMedianAge +
## sdrooms + sdbedrooms + population + medianIncome + sdbedrooms:population
##
## Df Sum of Sq RSS AIC
## + housingMedianAge:medianIncome 1 2.4285e+11 7.4611e+13 323128
## + sdrooms:sdbedrooms 1 2.2004e+11 7.4633e+13 323132
## + population:medianIncome 1 1.7223e+11 7.4681e+13 323142
## + latitude:sdrooms 1 1.3623e+11 7.4717e+13 323149
## + housingMedianAge:sdrooms 1 1.3522e+11 7.4718e+13 323149
## + sdrooms:medianIncome 1 1.2623e+11 7.4727e+13 323151
## + housingMedianAge:sdbedrooms 1 1.1572e+11 7.4738e+13 323153
## + latitude:medianIncome 1 9.4737e+10 7.4759e+13 323157
## + sdrooms:population 1 8.4212e+10 7.4769e+13 323159
## + latitude:housingMedianAge 1 6.7095e+10 7.4786e+13 323162
## + longitude:population 1 3.2672e+10 7.4821e+13 323169
## + longitude:housingMedianAge 1 3.0988e+10 7.4822e+13 323169
## + latitude:population 1 1.4350e+10 7.4839e+13 323172
## + sdbedrooms:medianIncome 1 1.1373e+10 7.4842e+13 323173
## <none> 7.4853e+13 323173
## + longitude:latitude 1 8.1370e+09 7.4845e+13 323173
## + latitude:sdbedrooms 1 6.4066e+09 7.4847e+13 323174
## + longitude:sdrooms 1 5.7826e+09 7.4848e+13 323174
## + longitude:medianIncome 1 2.8338e+09 7.4851e+13 323174
## + longitude:sdbedrooms 1 8.0890e+08 7.4853e+13 323175
## + housingMedianAge:population 1 4.7808e+08 7.4853e+13 323175
## - sdbedrooms:population 1 8.1291e+11 7.5666e+13 323327
## - sdrooms 1 1.2733e+12 7.6127e+13 323415
## - housingMedianAge 1 1.7377e+12 7.6591e+13 323503
## - longitude 1 1.2234e+13 8.7087e+13 325358
## - latitude 1 1.2420e+13 8.7273e+13 325389
## - medianIncome 1 3.9545e+13 1.1440e+14 329299
##
## Step: AIC=323128
## medianHouseValue ~ longitude + latitude + housingMedianAge +
## sdrooms + sdbedrooms + population + medianIncome + sdbedrooms:population +
## housingMedianAge:medianIncome
##
## Df Sum of Sq RSS AIC
## + housingMedianAge:sdrooms 1 4.3492e+11 7.4176e+13 323046
## + population:medianIncome 1 3.1782e+11 7.4293e+13 323068
## + sdrooms:sdbedrooms 1 2.1592e+11 7.4395e+13 323088
## + latitude:sdrooms 1 1.1523e+11 7.4495e+13 323108
## + latitude:medianIncome 1 1.1523e+11 7.4495e+13 323108
## + housingMedianAge:sdbedrooms 1 9.7461e+10 7.4513e+13 323111
## + sdrooms:medianIncome 1 8.8458e+10 7.4522e+13 323113
## + latitude:housingMedianAge 1 4.4707e+10 7.4566e+13 323121
## + sdrooms:population 1 4.1012e+10 7.4570e+13 323122
## + longitude:housingMedianAge 1 3.6588e+10 7.4574e+13 323123
## + longitude:population 1 3.3343e+10 7.4577e+13 323124
## + latitude:population 1 1.2704e+10 7.4598e+13 323128
## + longitude:medianIncome 1 1.2356e+10 7.4598e+13 323128
## <none> 7.4611e+13 323128
## + longitude:latitude 1 7.3623e+09 7.4603e+13 323129
## + latitude:sdbedrooms 1 5.0723e+09 7.4606e+13 323129
## + sdbedrooms:medianIncome 1 3.9077e+09 7.4607e+13 323129
## + longitude:sdrooms 1 1.2482e+09 7.4609e+13 323130
## + longitude:sdbedrooms 1 5.7306e+08 7.4610e+13 323130
## + housingMedianAge:population 1 5.3171e+08 7.4610e+13 323130
## - housingMedianAge:medianIncome 1 2.4285e+11 7.4853e+13 323173
## - sdbedrooms:population 1 7.8477e+11 7.5395e+13 323277
## - sdrooms 1 1.1647e+12 7.5775e+13 323350
## - longitude 1 1.2418e+13 8.7029e+13 325350
## - latitude 1 1.2616e+13 8.7226e+13 325383
##
## Step: AIC=323045.6
## medianHouseValue ~ longitude + latitude + housingMedianAge +
## sdrooms + sdbedrooms + population + medianIncome + sdbedrooms:population +
## housingMedianAge:medianIncome + housingMedianAge:sdrooms
##
## Df Sum of Sq RSS AIC
## + population:medianIncome 1 3.3293e+11 7.3843e+13 322983
## + housingMedianAge:sdbedrooms 1 2.3641e+11 7.3939e+13 323001
## + latitude:medianIncome 1 1.2852e+11 7.4047e+13 323023
## + sdrooms:medianIncome 1 9.0182e+10 7.4085e+13 323030
## + latitude:sdrooms 1 8.2306e+10 7.4093e+13 323032
## + longitude:housingMedianAge 1 4.2105e+10 7.4134e+13 323039
## + longitude:population 1 2.9871e+10 7.4146e+13 323042
## + sdrooms:population 1 2.5830e+10 7.4150e+13 323043
## + sdrooms:sdbedrooms 1 1.9001e+10 7.4157e+13 323044
## + latitude:housingMedianAge 1 1.7004e+10 7.4159e+13 323044
## + longitude:medianIncome 1 1.3484e+10 7.4162e+13 323045
## + latitude:population 1 1.3195e+10 7.4162e+13 323045
## <none> 7.4176e+13 323046
## + latitude:sdbedrooms 1 7.7548e+09 7.4168e+13 323046
## + longitude:sdrooms 1 4.4772e+09 7.4171e+13 323047
## + longitude:latitude 1 4.1972e+09 7.4171e+13 323047
## + longitude:sdbedrooms 1 2.9386e+09 7.4173e+13 323047
## + sdbedrooms:medianIncome 1 5.4457e+08 7.4175e+13 323047
## + housingMedianAge:population 1 4.1569e+08 7.4175e+13 323048
## - housingMedianAge:sdrooms 1 4.3492e+11 7.4611e+13 323128
## - sdbedrooms:population 1 5.1610e+11 7.4692e+13 323144
## - housingMedianAge:medianIncome 1 5.4255e+11 7.4718e+13 323149
## - longitude 1 1.2522e+13 8.6698e+13 325297
## - latitude 1 1.2682e+13 8.6858e+13 325324
##
## Step: AIC=322982.6
## medianHouseValue ~ longitude + latitude + housingMedianAge +
## sdrooms + sdbedrooms + population + medianIncome + sdbedrooms:population +
## housingMedianAge:medianIncome + housingMedianAge:sdrooms +
## population:medianIncome
##
## Df Sum of Sq RSS AIC
## + sdrooms:population 1 6.9735e+11 7.3145e+13 322848
## + housingMedianAge:sdbedrooms 1 2.4744e+11 7.3595e+13 322936
## + latitude:sdrooms 1 1.0346e+11 7.3739e+13 322964
## + latitude:medianIncome 1 9.5267e+10 7.3747e+13 322966
## + sdrooms:medianIncome 1 5.1295e+10 7.3791e+13 322975
## + longitude:housingMedianAge 1 3.8198e+10 7.3805e+13 322977
## + longitude:population 1 3.4387e+10 7.3808e+13 322978
## + latitude:population 1 2.1228e+10 7.3822e+13 322980
## + housingMedianAge:population 1 2.0344e+10 7.3822e+13 322981
## + latitude:housingMedianAge 1 2.0205e+10 7.3823e+13 322981
## + sdrooms:sdbedrooms 1 1.2420e+10 7.3830e+13 322982
## <none> 7.3843e+13 322983
## + longitude:sdrooms 1 8.1333e+09 7.3835e+13 322983
## + latitude:sdbedrooms 1 8.0072e+09 7.3835e+13 322983
## + sdbedrooms:medianIncome 1 5.9592e+09 7.3837e+13 322983
## + longitude:medianIncome 1 5.7057e+09 7.3837e+13 322983
## + longitude:latitude 1 5.6452e+09 7.3837e+13 322983
## + longitude:sdbedrooms 1 2.2722e+09 7.3840e+13 322984
## - population:medianIncome 1 3.3293e+11 7.4176e+13 323046
## - housingMedianAge:sdrooms 1 4.5003e+11 7.4293e+13 323068
## - sdbedrooms:population 1 5.1949e+11 7.4362e+13 323082
## - housingMedianAge:medianIncome 1 7.3111e+11 7.4574e+13 323123
## - longitude 1 1.2465e+13 8.6308e+13 325234
## - latitude 1 1.2676e+13 8.6519e+13 325269
##
## Step: AIC=322847.5
## medianHouseValue ~ longitude + latitude + housingMedianAge +
## sdrooms + sdbedrooms + population + medianIncome + sdbedrooms:population +
## housingMedianAge:medianIncome + housingMedianAge:sdrooms +
## population:medianIncome + sdrooms:population
##
## Df Sum of Sq RSS AIC
## + housingMedianAge:sdbedrooms 1 3.6900e+11 7.2776e+13 322776
## + latitude:medianIncome 1 1.0249e+11 7.3043e+13 322829
## + sdrooms:medianIncome 1 7.7354e+10 7.3068e+13 322834
## + latitude:population 1 5.1063e+10 7.3094e+13 322839
## + longitude:housingMedianAge 1 4.4974e+10 7.3100e+13 322841
## + latitude:sdrooms 1 3.6014e+10 7.3109e+13 322842
## + longitude:population 1 3.3522e+10 7.3112e+13 322843
## + latitude:housingMedianAge 1 2.2877e+10 7.3123e+13 322845
## + housingMedianAge:population 1 1.0942e+10 7.3134e+13 322847
## <none> 7.3145e+13 322848
## + longitude:medianIncome 1 8.5406e+09 7.3137e+13 322848
## + latitude:sdbedrooms 1 6.5210e+09 7.3139e+13 322848
## + sdbedrooms:medianIncome 1 4.5228e+09 7.3141e+13 322849
## + longitude:sdbedrooms 1 3.5103e+09 7.3142e+13 322849
## + longitude:latitude 1 2.1007e+09 7.3143e+13 322849
## + sdrooms:sdbedrooms 1 1.3361e+09 7.3144e+13 322849
## + longitude:sdrooms 1 7.4286e+08 7.3145e+13 322849
## - housingMedianAge:sdrooms 1 3.7132e+11 7.3517e+13 322919
## - sdrooms:population 1 6.9735e+11 7.3843e+13 322983
## - housingMedianAge:medianIncome 1 6.9972e+11 7.3845e+13 322983
## - population:medianIncome 1 1.0044e+12 7.4150e+13 323043
## - sdbedrooms:population 1 1.1673e+12 7.4313e+13 323074
## - longitude 1 1.1909e+13 8.5054e+13 325025
## - latitude 1 1.2030e+13 8.5175e+13 325045
##
## Step: AIC=322776.5
## medianHouseValue ~ longitude + latitude + housingMedianAge +
## sdrooms + sdbedrooms + population + medianIncome + sdbedrooms:population +
## housingMedianAge:medianIncome + housingMedianAge:sdrooms +
## population:medianIncome + sdrooms:population + housingMedianAge:sdbedrooms
##
## Df Sum of Sq RSS AIC
## + sdrooms:medianIncome 1 1.6701e+11 7.2609e+13 322745
## + sdrooms:sdbedrooms 1 1.1237e+11 7.2664e+13 322756
## + latitude:medianIncome 1 1.0996e+11 7.2666e+13 322757
## + longitude:housingMedianAge 1 5.5922e+10 7.2720e+13 322767
## + latitude:population 1 4.1357e+10 7.2735e+13 322770
## + longitude:population 1 3.1812e+10 7.2745e+13 322772
## + latitude:sdrooms 1 1.9715e+10 7.2757e+13 322775
## + housingMedianAge:population 1 1.4889e+10 7.2762e+13 322775
## <none> 7.2776e+13 322776
## + longitude:medianIncome 1 8.5064e+09 7.2768e+13 322777
## + latitude:housingMedianAge 1 8.3633e+09 7.2768e+13 322777
## + sdbedrooms:medianIncome 1 2.0666e+09 7.2774e+13 322778
## + latitude:sdbedrooms 1 1.5497e+09 7.2775e+13 322778
## + longitude:latitude 1 1.3878e+09 7.2775e+13 322778
## + longitude:sdbedrooms 1 7.9463e+08 7.2776e+13 322778
## + longitude:sdrooms 1 3.9618e+08 7.2776e+13 322778
## - housingMedianAge:sdbedrooms 1 3.6900e+11 7.3145e+13 322848
## - housingMedianAge:sdrooms 1 6.9765e+11 7.3474e+13 322912
## - sdrooms:population 1 8.1891e+11 7.3595e+13 322936
## - housingMedianAge:medianIncome 1 1.0647e+12 7.3841e+13 322984
## - population:medianIncome 1 1.1241e+12 7.3901e+13 322996
## - sdbedrooms:population 1 1.3303e+12 7.4107e+13 323036
## - longitude 1 1.1984e+13 8.4761e+13 324977
## - latitude 1 1.2105e+13 8.4881e+13 324997
##
## Step: AIC=322745.2
## medianHouseValue ~ longitude + latitude + housingMedianAge +
## sdrooms + sdbedrooms + population + medianIncome + sdbedrooms:population +
## housingMedianAge:medianIncome + housingMedianAge:sdrooms +
## population:medianIncome + sdrooms:population + housingMedianAge:sdbedrooms +
## sdrooms:medianIncome
##
## Df Sum of Sq RSS AIC
## + latitude:medianIncome 1 1.0230e+11 7.2507e+13 322727
## + sdbedrooms:medianIncome 1 6.8267e+10 7.2541e+13 322734
## + sdrooms:sdbedrooms 1 5.9650e+10 7.2550e+13 322735
## + longitude:housingMedianAge 1 5.8211e+10 7.2551e+13 322736
## + latitude:population 1 3.9698e+10 7.2570e+13 322739
## + longitude:population 1 2.9332e+10 7.2580e+13 322741
## + housingMedianAge:population 1 1.5636e+10 7.2594e+13 322744
## + latitude:sdrooms 1 1.4754e+10 7.2595e+13 322744
## <none> 7.2609e+13 322745
## + latitude:housingMedianAge 1 6.3088e+09 7.2603e+13 322746
## + longitude:medianIncome 1 5.8863e+09 7.2603e+13 322746
## + longitude:latitude 1 9.9698e+08 7.2608e+13 322747
## + longitude:sdrooms 1 5.5443e+08 7.2609e+13 322747
## + latitude:sdbedrooms 1 1.6948e+08 7.2609e+13 322747
## + longitude:sdbedrooms 1 1.0961e+08 7.2609e+13 322747
## - sdrooms:medianIncome 1 1.6701e+11 7.2776e+13 322776
## - housingMedianAge:sdbedrooms 1 4.5866e+11 7.3068e+13 322834
## - housingMedianAge:sdrooms 1 7.9748e+11 7.3407e+13 322901
## - sdrooms:population 1 8.7848e+11 7.3488e+13 322917
## - housingMedianAge:medianIncome 1 1.0819e+12 7.3691e+13 322957
## - population:medianIncome 1 1.1003e+12 7.3710e+13 322961
## - sdbedrooms:population 1 1.3837e+12 7.3993e+13 323016
## - longitude 1 1.1810e+13 8.4419e+13 324920
## - latitude 1 1.1961e+13 8.4570e+13 324946
##
## Step: AIC=322726.9
## medianHouseValue ~ longitude + latitude + housingMedianAge +
## sdrooms + sdbedrooms + population + medianIncome + sdbedrooms:population +
## housingMedianAge:medianIncome + housingMedianAge:sdrooms +
## population:medianIncome + sdrooms:population + housingMedianAge:sdbedrooms +
## sdrooms:medianIncome + latitude:medianIncome
##
## Df Sum of Sq RSS AIC
## + longitude:medianIncome 1 5.8314e+11 7.1924e+13 322612
## + latitude:sdrooms 1 7.5928e+10 7.2431e+13 322714
## + sdbedrooms:medianIncome 1 7.0675e+10 7.2436e+13 322715
## + sdrooms:sdbedrooms 1 5.6587e+10 7.2450e+13 322718
## + latitude:population 1 4.5301e+10 7.2462e+13 322720
## + longitude:housingMedianAge 1 4.1422e+10 7.2466e+13 322721
## + longitude:sdrooms 1 3.5660e+10 7.2471e+13 322722
## + longitude:population 1 3.2731e+10 7.2474e+13 322722
## + latitude:housingMedianAge 1 1.6782e+10 7.2490e+13 322726
## + housingMedianAge:population 1 1.4654e+10 7.2492e+13 322726
## <none> 7.2507e+13 322727
## + longitude:latitude 1 3.8096e+09 7.2503e+13 322728
## + longitude:sdbedrooms 1 8.7677e+08 7.2506e+13 322729
## + latitude:sdbedrooms 1 1.2961e+08 7.2507e+13 322729
## - latitude:medianIncome 1 1.0230e+11 7.2609e+13 322745
## - sdrooms:medianIncome 1 1.5936e+11 7.2666e+13 322757
## - housingMedianAge:sdbedrooms 1 4.6404e+11 7.2971e+13 322817
## - housingMedianAge:sdrooms 1 8.1082e+11 7.3318e+13 322886
## - sdrooms:population 1 8.8636e+11 7.3393e+13 322900
## - population:medianIncome 1 1.0655e+12 7.3573e+13 322936
## - housingMedianAge:medianIncome 1 1.1106e+12 7.3618e+13 322944
## - sdbedrooms:population 1 1.3893e+12 7.3896e+13 322999
## - longitude 1 1.1773e+13 8.4280e+13 324899
##
## Step: AIC=322612.2
## medianHouseValue ~ longitude + latitude + housingMedianAge +
## sdrooms + sdbedrooms + population + medianIncome + sdbedrooms:population +
## housingMedianAge:medianIncome + housingMedianAge:sdrooms +
## population:medianIncome + sdrooms:population + housingMedianAge:sdbedrooms +
## sdrooms:medianIncome + latitude:medianIncome + longitude:medianIncome
##
## Df Sum of Sq RSS AIC
## + sdbedrooms:medianIncome 1 2.2846e+11 7.1695e+13 322568
## + latitude:sdrooms 1 8.4280e+10 7.1840e+13 322597
## + latitude:population 1 5.3824e+10 7.1870e+13 322603
## + longitude:housingMedianAge 1 4.5212e+10 7.1879e+13 322605
## + longitude:population 1 3.4666e+10 7.1889e+13 322607
## + sdrooms:sdbedrooms 1 3.2789e+10 7.1891e+13 322608
## + latitude:housingMedianAge 1 2.8940e+10 7.1895e+13 322608
## + housingMedianAge:population 1 2.0618e+10 7.1903e+13 322610
## <none> 7.1924e+13 322612
## + longitude:sdrooms 1 5.3600e+09 7.1919e+13 322613
## + longitude:latitude 1 5.1196e+09 7.1919e+13 322613
## + longitude:sdbedrooms 1 1.4356e+08 7.1924e+13 322614
## + latitude:sdbedrooms 1 7.2701e+07 7.1924e+13 322614
## - sdrooms:medianIncome 1 1.8091e+11 7.2105e+13 322647
## - housingMedianAge:sdbedrooms 1 5.3356e+11 7.2458e+13 322717
## - longitude:medianIncome 1 5.8314e+11 7.2507e+13 322727
## - latitude:medianIncome 1 6.7955e+11 7.2603e+13 322746
## - sdrooms:population 1 8.6791e+11 7.2792e+13 322784
## - housingMedianAge:sdrooms 1 9.3376e+11 7.2858e+13 322797
## - housingMedianAge:medianIncome 1 9.9064e+11 7.2915e+13 322808
## - population:medianIncome 1 9.9602e+11 7.2920e+13 322809
## - sdbedrooms:population 1 1.3228e+12 7.3247e+13 322874
##
## Step: AIC=322568.3
## medianHouseValue ~ longitude + latitude + housingMedianAge +
## sdrooms + sdbedrooms + population + medianIncome + sdbedrooms:population +
## housingMedianAge:medianIncome + housingMedianAge:sdrooms +
## population:medianIncome + sdrooms:population + housingMedianAge:sdbedrooms +
## sdrooms:medianIncome + latitude:medianIncome + longitude:medianIncome +
## sdbedrooms:medianIncome
##
## Df Sum of Sq RSS AIC
## + sdrooms:sdbedrooms 1 1.9157e+11 7.1504e+13 322532
## + latitude:sdrooms 1 8.7846e+10 7.1608e+13 322553
## + latitude:population 1 4.7492e+10 7.1648e+13 322561
## + longitude:housingMedianAge 1 4.1863e+10 7.1654e+13 322562
## + longitude:population 1 3.1261e+10 7.1664e+13 322564
## + latitude:housingMedianAge 1 3.0549e+10 7.1665e+13 322564
## + housingMedianAge:population 1 1.8430e+10 7.1677e+13 322567
## + longitude:sdrooms 1 1.4656e+10 7.1681e+13 322567
## <none> 7.1695e+13 322568
## + longitude:latitude 1 3.5247e+09 7.1692e+13 322570
## + longitude:sdbedrooms 1 1.0683e+09 7.1694e+13 322570
## + latitude:sdbedrooms 1 6.6990e+06 7.1695e+13 322570
## - sdbedrooms:medianIncome 1 2.2846e+11 7.1924e+13 322612
## - sdrooms:medianIncome 1 3.9905e+11 7.2095e+13 322646
## - housingMedianAge:sdbedrooms 1 4.9029e+11 7.2186e+13 322665
## - longitude:medianIncome 1 7.4092e+11 7.2436e+13 322715
## - latitude:medianIncome 1 8.4200e+11 7.2537e+13 322735
## - sdrooms:population 1 8.8630e+11 7.2582e+13 322744
## - housingMedianAge:sdrooms 1 9.7553e+11 7.2671e+13 322762
## - housingMedianAge:medianIncome 1 9.9630e+11 7.2692e+13 322766
## - population:medianIncome 1 1.0097e+12 7.2705e+13 322768
## - sdbedrooms:population 1 1.3741e+12 7.3070e+13 322841
##
## Step: AIC=322531.6
## medianHouseValue ~ longitude + latitude + housingMedianAge +
## sdrooms + sdbedrooms + population + medianIncome + sdbedrooms:population +
## housingMedianAge:medianIncome + housingMedianAge:sdrooms +
## population:medianIncome + sdrooms:population + housingMedianAge:sdbedrooms +
## sdrooms:medianIncome + latitude:medianIncome + longitude:medianIncome +
## sdbedrooms:medianIncome + sdrooms:sdbedrooms
##
## Df Sum of Sq RSS AIC
## + latitude:sdrooms 1 1.6546e+11 7.1338e+13 322500
## + latitude:population 1 4.7931e+10 7.1456e+13 322524
## + longitude:housingMedianAge 1 4.4808e+10 7.1459e+13 322525
## + latitude:housingMedianAge 1 2.8895e+10 7.1475e+13 322528
## + longitude:population 1 2.6342e+10 7.1478e+13 322528
## + longitude:sdrooms 1 2.4411e+10 7.1479e+13 322529
## + housingMedianAge:population 1 2.0701e+10 7.1483e+13 322529
## + latitude:sdbedrooms 1 1.8144e+10 7.1486e+13 322530
## <none> 7.1504e+13 322532
## + longitude:sdbedrooms 1 5.9711e+09 7.1498e+13 322532
## + longitude:latitude 1 3.2395e+09 7.1501e+13 322533
## - sdrooms:sdbedrooms 1 1.9157e+11 7.1695e+13 322568
## - sdbedrooms:medianIncome 1 3.8725e+11 7.1891e+13 322608
## - sdrooms:medianIncome 1 4.4968e+11 7.1954e+13 322620
## - housingMedianAge:sdbedrooms 1 6.6996e+11 7.2174e+13 322664
## - longitude:medianIncome 1 7.7405e+11 7.2278e+13 322685
## - sdrooms:population 1 8.3708e+11 7.2341e+13 322698
## - sdbedrooms:population 1 8.6283e+11 7.2367e+13 322703
## - latitude:medianIncome 1 8.7242e+11 7.2376e+13 322705
## - population:medianIncome 1 9.6774e+11 7.2472e+13 322724
## - housingMedianAge:sdrooms 1 1.0054e+12 7.2509e+13 322731
## - housingMedianAge:medianIncome 1 1.0540e+12 7.2558e+13 322741
##
## Step: AIC=322500.1
## medianHouseValue ~ longitude + latitude + housingMedianAge +
## sdrooms + sdbedrooms + population + medianIncome + sdbedrooms:population +
## housingMedianAge:medianIncome + housingMedianAge:sdrooms +
## population:medianIncome + sdrooms:population + housingMedianAge:sdbedrooms +
## sdrooms:medianIncome + latitude:medianIncome + longitude:medianIncome +
## sdbedrooms:medianIncome + sdrooms:sdbedrooms + latitude:sdrooms
##
## Df Sum of Sq RSS AIC
## + longitude:sdrooms 1 1.6730e+11 7.1171e+13 322468
## + latitude:sdbedrooms 1 1.5359e+11 7.1185e+13 322471
## + longitude:sdbedrooms 1 1.0590e+11 7.1233e+13 322481
## + longitude:housingMedianAge 1 7.1354e+10 7.1267e+13 322488
## + latitude:population 1 6.2838e+10 7.1276e+13 322489
## + longitude:population 1 3.7075e+10 7.1301e+13 322495
## + housingMedianAge:population 1 1.5212e+10 7.1323e+13 322499
## + latitude:housingMedianAge 1 1.4640e+10 7.1324e+13 322499
## <none> 7.1338e+13 322500
## + longitude:latitude 1 3.1582e+09 7.1335e+13 322501
## - latitude:sdrooms 1 1.6546e+11 7.1504e+13 322532
## - sdrooms:sdbedrooms 1 2.6919e+11 7.1608e+13 322553
## - sdrooms:medianIncome 1 4.3881e+11 7.1777e+13 322587
## - sdbedrooms:medianIncome 1 4.4187e+11 7.1780e+13 322587
## - sdrooms:population 1 6.6352e+11 7.2002e+13 322632
## - housingMedianAge:sdbedrooms 1 6.7687e+11 7.2015e+13 322635
## - longitude:medianIncome 1 7.9703e+11 7.2135e+13 322659
## - population:medianIncome 1 8.5497e+11 7.2193e+13 322670
## - housingMedianAge:sdrooms 1 9.1178e+11 7.2250e+13 322682
## - sdbedrooms:population 1 9.5302e+11 7.2291e+13 322690
## - latitude:medianIncome 1 9.7488e+11 7.2313e+13 322694
## - housingMedianAge:medianIncome 1 9.7646e+11 7.2315e+13 322695
##
## Step: AIC=322468.2
## medianHouseValue ~ longitude + latitude + housingMedianAge +
## sdrooms + sdbedrooms + population + medianIncome + sdbedrooms:population +
## housingMedianAge:medianIncome + housingMedianAge:sdrooms +
## population:medianIncome + sdrooms:population + housingMedianAge:sdbedrooms +
## sdrooms:medianIncome + latitude:medianIncome + longitude:medianIncome +
## sdbedrooms:medianIncome + sdrooms:sdbedrooms + latitude:sdrooms +
## longitude:sdrooms
##
## Df Sum of Sq RSS AIC
## + latitude:sdbedrooms 1 1.4258e+11 7.1029e+13 322441
## + latitude:population 1 5.5689e+10 7.1115e+13 322459
## + longitude:housingMedianAge 1 5.2057e+10 7.1119e+13 322460
## + longitude:population 1 2.9523e+10 7.1142e+13 322464
## + latitude:housingMedianAge 1 2.5354e+10 7.1146e+13 322465
## + longitude:sdbedrooms 1 1.5083e+10 7.1156e+13 322467
## + housingMedianAge:population 1 1.1258e+10 7.1160e+13 322468
## <none> 7.1171e+13 322468
## + longitude:latitude 1 5.6774e+08 7.1171e+13 322470
## - longitude:sdrooms 1 1.6730e+11 7.1338e+13 322500
## - latitude:sdrooms 1 3.0835e+11 7.1479e+13 322529
## - sdrooms:medianIncome 1 3.4541e+11 7.1517e+13 322536
## - sdrooms:sdbedrooms 1 3.6297e+11 7.1534e+13 322540
## - sdbedrooms:medianIncome 1 4.1468e+11 7.1586e+13 322550
## - sdrooms:population 1 5.1633e+11 7.1687e+13 322571
## - housingMedianAge:sdbedrooms 1 6.7090e+11 7.1842e+13 322602
## - housingMedianAge:sdrooms 1 6.9138e+11 7.1863e+13 322606
## - population:medianIncome 1 7.5653e+11 7.1928e+13 322619
## - housingMedianAge:medianIncome 1 8.4111e+11 7.2012e+13 322636
## - sdbedrooms:population 1 8.5465e+11 7.2026e+13 322639
## - longitude:medianIncome 1 9.4789e+11 7.2119e+13 322657
## - latitude:medianIncome 1 1.1069e+12 7.2278e+13 322689
##
## Step: AIC=322441.2
## medianHouseValue ~ longitude + latitude + housingMedianAge +
## sdrooms + sdbedrooms + population + medianIncome + sdbedrooms:population +
## housingMedianAge:medianIncome + housingMedianAge:sdrooms +
## population:medianIncome + sdrooms:population + housingMedianAge:sdbedrooms +
## sdrooms:medianIncome + latitude:medianIncome + longitude:medianIncome +
## sdbedrooms:medianIncome + sdrooms:sdbedrooms + latitude:sdrooms +
## longitude:sdrooms + latitude:sdbedrooms
##
## Df Sum of Sq RSS AIC
## + longitude:sdbedrooms 1 2.4238e+11 7.0786e+13 322394
## + longitude:housingMedianAge 1 6.0539e+10 7.0968e+13 322431
## + latitude:population 1 4.6522e+10 7.0982e+13 322434
## + longitude:population 1 2.3524e+10 7.1005e+13 322438
## + latitude:housingMedianAge 1 2.1757e+10 7.1007e+13 322439
## <none> 7.1029e+13 322441
## + housingMedianAge:population 1 8.9514e+09 7.1020e+13 322441
## + longitude:latitude 1 1.0160e+09 7.1028e+13 322443
## - latitude:sdbedrooms 1 1.4258e+11 7.1171e+13 322468
## - longitude:sdrooms 1 1.5629e+11 7.1185e+13 322471
## - sdrooms:sdbedrooms 1 2.6007e+11 7.1289e+13 322492
## - sdrooms:medianIncome 1 3.4440e+11 7.1373e+13 322509
## - sdbedrooms:medianIncome 1 3.6530e+11 7.1394e+13 322513
## - sdrooms:population 1 3.8222e+11 7.1411e+13 322517
## - latitude:sdrooms 1 4.5021e+11 7.1479e+13 322531
## - sdbedrooms:population 1 6.0459e+11 7.1633e+13 322562
## - housingMedianAge:sdbedrooms 1 6.1806e+11 7.1647e+13 322564
## - population:medianIncome 1 6.4437e+11 7.1673e+13 322570
## - housingMedianAge:sdrooms 1 6.4798e+11 7.1677e+13 322570
## - housingMedianAge:medianIncome 1 7.8913e+11 7.1818e+13 322599
## - longitude:medianIncome 1 9.4882e+11 7.1977e+13 322631
## - latitude:medianIncome 1 1.2369e+12 7.2265e+13 322689
##
## Step: AIC=322393.9
## medianHouseValue ~ longitude + latitude + housingMedianAge +
## sdrooms + sdbedrooms + population + medianIncome + sdbedrooms:population +
## housingMedianAge:medianIncome + housingMedianAge:sdrooms +
## population:medianIncome + sdrooms:population + housingMedianAge:sdbedrooms +
## sdrooms:medianIncome + latitude:medianIncome + longitude:medianIncome +
## sdbedrooms:medianIncome + sdrooms:sdbedrooms + latitude:sdrooms +
## longitude:sdrooms + latitude:sdbedrooms + longitude:sdbedrooms
##
## Df Sum of Sq RSS AIC
## + longitude:housingMedianAge 1 5.9960e+10 7.0726e+13 322384
## + latitude:population 1 4.7425e+10 7.0739e+13 322386
## + longitude:population 1 2.8292e+10 7.0758e+13 322390
## + latitude:housingMedianAge 1 2.4302e+10 7.0762e+13 322391
## <none> 7.0786e+13 322394
## + housingMedianAge:population 1 5.1583e+09 7.0781e+13 322395
## + longitude:latitude 1 9.8818e+08 7.0785e+13 322396
## - sdrooms:sdbedrooms 1 1.0609e+11 7.0892e+13 322413
## - longitude:sdbedrooms 1 2.4238e+11 7.1029e+13 322441
## - sdrooms:population 1 2.7394e+11 7.1060e+13 322448
## - sdbedrooms:medianIncome 1 3.0933e+11 7.1096e+13 322455
## - sdrooms:medianIncome 1 3.2410e+11 7.1110e+13 322458
## - latitude:sdbedrooms 1 3.6988e+11 7.1156e+13 322467
## - housingMedianAge:sdbedrooms 1 3.9157e+11 7.1178e+13 322472
## - longitude:sdrooms 1 3.9741e+11 7.1184e+13 322473
## - housingMedianAge:sdrooms 1 4.2407e+11 7.1210e+13 322478
## - sdbedrooms:population 1 4.3506e+11 7.1221e+13 322480
## - population:medianIncome 1 5.4506e+11 7.1331e+13 322503
## - housingMedianAge:medianIncome 1 5.8223e+11 7.1368e+13 322510
## - latitude:sdrooms 1 6.6456e+11 7.1451e+13 322527
## - longitude:medianIncome 1 1.1904e+12 7.1977e+13 322633
## - latitude:medianIncome 1 1.4773e+12 7.2263e+13 322690
##
## Step: AIC=322383.6
## medianHouseValue ~ longitude + latitude + housingMedianAge +
## sdrooms + sdbedrooms + population + medianIncome + sdbedrooms:population +
## housingMedianAge:medianIncome + housingMedianAge:sdrooms +
## population:medianIncome + sdrooms:population + housingMedianAge:sdbedrooms +
## sdrooms:medianIncome + latitude:medianIncome + longitude:medianIncome +
## sdbedrooms:medianIncome + sdrooms:sdbedrooms + latitude:sdrooms +
## longitude:sdrooms + latitude:sdbedrooms + longitude:sdbedrooms +
## longitude:housingMedianAge
##
## Df Sum of Sq RSS AIC
## + latitude:housingMedianAge 1 1.4951e+12 6.9231e+13 322077
## + latitude:population 1 8.8385e+10 7.0638e+13 322368
## + longitude:population 1 6.3162e+10 7.0663e+13 322373
## <none> 7.0726e+13 322384
## + longitude:latitude 1 5.3429e+09 7.0721e+13 322385
## + housingMedianAge:population 1 2.3739e+09 7.0724e+13 322385
## - longitude:housingMedianAge 1 5.9960e+10 7.0786e+13 322394
## - sdrooms:sdbedrooms 1 1.0781e+11 7.0834e+13 322404
## - longitude:sdbedrooms 1 2.4180e+11 7.0968e+13 322431
## - sdrooms:population 1 2.7524e+11 7.1001e+13 322438
## - sdbedrooms:medianIncome 1 3.1137e+11 7.1038e+13 322445
## - sdrooms:medianIncome 1 3.2859e+11 7.1055e+13 322449
## - latitude:sdbedrooms 1 3.7525e+11 7.1101e+13 322458
## - longitude:sdrooms 1 3.7769e+11 7.1104e+13 322459
## - housingMedianAge:sdbedrooms 1 4.0406e+11 7.1130e+13 322464
## - sdbedrooms:population 1 4.1582e+11 7.1142e+13 322466
## - housingMedianAge:sdrooms 1 4.3856e+11 7.1165e+13 322471
## - population:medianIncome 1 5.4478e+11 7.1271e+13 322492
## - housingMedianAge:medianIncome 1 5.9754e+11 7.1324e+13 322503
## - latitude:sdrooms 1 6.6664e+11 7.1393e+13 322517
## - longitude:medianIncome 1 1.1814e+12 7.1908e+13 322621
## - latitude:medianIncome 1 1.4676e+12 7.2194e+13 322678
##
## Step: AIC=322077
## medianHouseValue ~ longitude + latitude + housingMedianAge +
## sdrooms + sdbedrooms + population + medianIncome + sdbedrooms:population +
## housingMedianAge:medianIncome + housingMedianAge:sdrooms +
## population:medianIncome + sdrooms:population + housingMedianAge:sdbedrooms +
## sdrooms:medianIncome + latitude:medianIncome + longitude:medianIncome +
## sdbedrooms:medianIncome + sdrooms:sdbedrooms + latitude:sdrooms +
## longitude:sdrooms + latitude:sdbedrooms + longitude:sdbedrooms +
## longitude:housingMedianAge + latitude:housingMedianAge
##
## Df Sum of Sq RSS AIC
## + longitude:population 1 5.1817e+10 6.9179e+13 322068
## + latitude:population 1 3.8993e+10 6.9192e+13 322071
## <none> 6.9231e+13 322077
## + longitude:latitude 1 5.4419e+09 6.9226e+13 322078
## + housingMedianAge:population 1 3.7030e+09 6.9227e+13 322078
## - housingMedianAge:sdrooms 1 6.3384e+10 6.9295e+13 322088
## - housingMedianAge:medianIncome 1 6.9735e+10 6.9301e+13 322089
## - sdrooms:sdbedrooms 1 9.9319e+10 6.9330e+13 322096
## - housingMedianAge:sdbedrooms 1 1.7033e+11 6.9401e+13 322110
## - sdrooms:medianIncome 1 2.5537e+11 6.9487e+13 322128
## - sdbedrooms:medianIncome 1 2.8560e+11 6.9517e+13 322134
## - longitude:sdbedrooms 1 3.0015e+11 6.9531e+13 322137
## - sdrooms:population 1 3.2885e+11 6.9560e+13 322143
## - sdbedrooms:population 1 4.1312e+11 6.9644e+13 322161
## - longitude:sdrooms 1 4.3118e+11 6.9662e+13 322165
## - latitude:sdbedrooms 1 4.6833e+11 6.9699e+13 322172
## - population:medianIncome 1 6.0065e+11 6.9832e+13 322200
## - latitude:sdrooms 1 7.6898e+11 7.0000e+13 322235
## - latitude:housingMedianAge 1 1.4951e+12 7.0726e+13 322384
## - longitude:housingMedianAge 1 1.5307e+12 7.0762e+13 322391
## - longitude:medianIncome 1 1.6641e+12 7.0895e+13 322418
## - latitude:medianIncome 1 2.0801e+12 7.1311e+13 322503
##
## Step: AIC=322068.1
## medianHouseValue ~ longitude + latitude + housingMedianAge +
## sdrooms + sdbedrooms + population + medianIncome + sdbedrooms:population +
## housingMedianAge:medianIncome + housingMedianAge:sdrooms +
## population:medianIncome + sdrooms:population + housingMedianAge:sdbedrooms +
## sdrooms:medianIncome + latitude:medianIncome + longitude:medianIncome +
## sdbedrooms:medianIncome + sdrooms:sdbedrooms + latitude:sdrooms +
## longitude:sdrooms + latitude:sdbedrooms + longitude:sdbedrooms +
## longitude:housingMedianAge + latitude:housingMedianAge +
## longitude:population
##
## Df Sum of Sq RSS AIC
## <none> 6.9179e+13 322068
## + housingMedianAge:population 1 8.5757e+09 6.9171e+13 322068
## + longitude:latitude 1 4.2922e+09 6.9175e+13 322069
## + latitude:population 1 1.4191e+09 6.9178e+13 322070
## - longitude:population 1 5.1817e+10 6.9231e+13 322077
## - housingMedianAge:sdrooms 1 6.3810e+10 6.9243e+13 322079
## - housingMedianAge:medianIncome 1 7.1310e+10 6.9251e+13 322081
## - sdrooms:sdbedrooms 1 9.5955e+10 6.9275e+13 322086
## - housingMedianAge:sdbedrooms 1 1.6868e+11 6.9348e+13 322101
## - sdrooms:medianIncome 1 2.5109e+11 6.9430e+13 322118
## - sdbedrooms:medianIncome 1 2.8128e+11 6.9461e+13 322125
## - longitude:sdbedrooms 1 3.0733e+11 6.9487e+13 322130
## - sdrooms:population 1 3.2852e+11 6.9508e+13 322135
## - longitude:sdrooms 1 4.2387e+11 6.9603e+13 322154
## - sdbedrooms:population 1 4.3037e+11 6.9610e+13 322156
## - latitude:sdbedrooms 1 4.7138e+11 6.9651e+13 322164
## - population:medianIncome 1 6.0489e+11 6.9784e+13 322192
## - latitude:sdrooms 1 7.6823e+11 6.9948e+13 322226
## - latitude:housingMedianAge 1 1.4837e+12 7.0663e+13 322373
## - longitude:housingMedianAge 1 1.5659e+12 7.0745e+13 322389
## - longitude:medianIncome 1 1.6609e+12 7.0840e+13 322409
## - latitude:medianIncome 1 2.0798e+12 7.1259e+13 322494##
## Call:
## lm(formula = medianHouseValue ~ longitude + latitude + housingMedianAge +
## sdrooms + sdbedrooms + population + medianIncome + sdbedrooms:population +
## housingMedianAge:medianIncome + housingMedianAge:sdrooms +
## population:medianIncome + sdrooms:population + housingMedianAge:sdbedrooms +
## sdrooms:medianIncome + latitude:medianIncome + longitude:medianIncome +
## sdbedrooms:medianIncome + sdrooms:sdbedrooms + latitude:sdrooms +
## longitude:sdrooms + latitude:sdbedrooms + longitude:sdbedrooms +
## longitude:housingMedianAge + latitude:housingMedianAge +
## longitude:population, data = cahousing_train)
##
## Coefficients:
## (Intercept) longitude
## 2.385e+06 2.767e+04
## latitude housingMedianAge
## 2.750e+04 -1.114e+05
## sdrooms sdbedrooms
## 5.356e+05 -2.596e+06
## population medianIncome
## -1.482e+02 -9.369e+05
## sdbedrooms:population housingMedianAge:medianIncome
## 4.584e+01 1.497e+02
## housingMedianAge:sdrooms population:medianIncome
## -2.130e+02 4.931e+00
## sdrooms:population housingMedianAge:sdbedrooms
## -5.177e+00 1.814e+03
## sdrooms:medianIncome latitude:medianIncome
## -1.356e+03 -1.319e+04
## longitude:medianIncome sdbedrooms:medianIncome
## -1.203e+04 1.109e+04
## sdrooms:sdbedrooms latitude:sdrooms
## -2.180e+02 7.886e+03
## longitude:sdrooms latitude:sdbedrooms
## 6.813e+03 -3.511e+04
## longitude:sdbedrooms longitude:housingMedianAge
## -3.197e+04 -1.302e+03
## latitude:housingMedianAge longitude:population
## -1.261e+03 -8.991e-01## n= 14447
##
## node), split, n, deviance, yval
## * denotes terminal node
##
## 1) root 14447 1.913508e+14 206632.40
## 2) medianIncome< 5.03515 11342 9.436258e+13 173093.80
## 4) medianIncome< 3.13565 5733 3.282567e+13 137158.30
## 8) sdrooms>=4.335204 3334 1.367816e+13 118547.40
## 16) medianIncome< 2.4187 1586 4.373468e+12 96168.10
## 32) latitude>=35.305 1056 1.680038e+12 86460.70
## 64) longitude>=-119.9 277 1.197040e+11 65480.87
## 128) latitude< 36.945 270 9.304502e+10 64187.41
## 256) sdrooms< 5.684405 241 6.593048e+10 61987.97
## 512) housingMedianAge>=10.5 232 3.903341e+10 61221.55
## 1024) medianIncome< 1.6147 60 2.666902e+09 52938.33 *
## 1025) medianIncome>=1.6147 172 3.081375e+10 64111.05
## 2050) population>=583.5 158 2.022190e+10 62659.49
## 4100) medianIncome< 2.10945 87 7.349080e+09 59464.37 *
## 4101) medianIncome>=2.10945 71 1.089633e+10 66574.65 *
## 2051) population< 583.5 14 6.501849e+09 80492.86 *
## 513) housingMedianAge< 10.5 9 2.324792e+10 81744.44 *
## 257) sdrooms>=5.684405 29 1.626011e+10 82465.52
## 514) latitude< 35.69 7 1.380357e+09 60642.86 *
## 515) latitude>=35.69 22 1.048546e+10 89409.09
## 1030) medianIncome>=2.3125 7 1.149974e+09 73571.43 *
## 1031) medianIncome< 2.3125 15 6.760280e+09 96800.00 *
## 129) latitude>=36.945 7 8.783754e+09 115371.40 *
## 65) longitude< -119.9 779 1.395058e+12 93920.79
## 130) latitude>=39.345 242 1.115291e+11 74867.76
## 260) housingMedianAge>=18.5 169 6.243163e+10 70851.47
## 520) sdbedrooms>=1.001548 149 3.562212e+10 68480.53
## 1040) medianIncome< 1.38725 10 3.372951e+09 50129.90 *
## 1041) medianIncome>=1.38725 139 2.863945e+10 69800.72
## 2082) longitude>=-123.725 109 1.956892e+10 67610.09
## 4164) latitude>=40.945 16 1.159618e+09 55487.50 *
## 4165) latitude< 40.945 93 1.565346e+10 69695.70
## 8330) housingMedianAge>=28.5 37 3.138052e+09 63951.35 *
## 8331) housingMedianAge< 28.5 56 1.048783e+10 73491.07 *
## 2083) longitude< -123.725 30 6.646952e+09 77760.00 *
## 521) sdbedrooms< 1.001548 20 1.973193e+10 88515.00
## 1042) population>=1031 7 9.030143e+08 67728.57 *
## 1043) population< 1031 13 1.417579e+10 99707.69 *
## 261) housingMedianAge< 18.5 73 4.006040e+10 84165.74
## 522) longitude< -122.02 42 1.773879e+10 77097.60
## 1044) medianIncome< 1.875 11 3.977313e+09 60827.18 *
## 1045) medianIncome>=1.875 31 9.816204e+09 82870.97 *
## 523) longitude>=-122.02 31 1.738054e+10 93741.94
## 1046) population>=724.5 21 4.417527e+09 85466.67 *
## 1047) population< 724.5 10 8.504956e+09 111120.00 *
## 131) latitude< 39.345 537 1.156088e+12 102507.10
## 262) longitude>=-121.73 334 5.188947e+11 92144.91
## 524) sdbedrooms>=0.9317544 326 3.550271e+11 90409.20
## 1048) sdbedrooms< 1.359339 300 2.637466e+11 87095.67
## 2096) housingMedianAge>=17.5 251 1.644577e+11 81594.02
## 4192) medianIncome< 1.98785 111 5.270381e+10 72202.70
## 8384) housingMedianAge< 44.5 98 3.577830e+10 69364.29
## 16768) medianIncome< 1.4979 26 2.411250e+09 59096.15 *
## 16769) medianIncome>=1.4979 72 2.963584e+10 73072.22
## 33538) housingMedianAge>=37.5 14 2.962493e+08 58892.86 *
## 33539) housingMedianAge< 37.5 58 2.584541e+10 76494.83
## 67078) population>=731.5 43 1.217828e+10 72374.42 *
## 67079) population< 731.5 15 1.084429e+10 88306.67 *
## 8385) housingMedianAge>=44.5 13 1.018398e+10 93600.00 *
## 4193) medianIncome>=1.98785 140 9.420208e+10 89040.00
## 8386) sdrooms< 5.258571 98 4.567622e+10 82856.12
## 16772) medianIncome>=2.02875 89 3.188179e+10 80908.99
## 33544) latitude< 37.98 46 9.408173e+09 74756.52
## 67088) longitude< -121.205 11 3.414473e+08 64154.55 *
## 67089) longitude>=-121.205 35 7.441715e+09 78088.57
## 134178) longitude>=-120.725 16 6.846194e+08 68156.25 *
## 134179) longitude< -120.725 19 3.849487e+09 86452.63 *
## 33545) latitude>=37.98 43 1.886968e+10 87490.70
## 67090) longitude< -121.355 27 9.709312e+09 78625.93
## 134180) latitude>=38.625 11 6.612000e+08 65200.00 *
## 134181) latitude< 38.625 16 5.702119e+09 87856.25 *
## 67091) longitude>=-121.355 16 3.458100e+09 102450.00 *
## 16773) medianIncome< 2.02875 9 1.012021e+10 102111.10 *
## 8387) sdrooms>=5.258571 42 3.603401e+10 103469.00
## 16774) sdbedrooms>=1.038293 32 2.255709e+10 97218.75
## 33548) medianIncome< 2.1922 15 3.215709e+09 85806.67 *
## 33549) medianIncome>=2.1922 17 1.566414e+10 107288.20 *
## 16775) sdbedrooms< 1.038293 10 8.226421e+09 123470.00 *
## 2097) housingMedianAge< 17.5 49 5.277495e+10 115277.60
## 4194) latitude< 37.45 13 1.128621e+10 87953.85 *
## 4195) latitude>=37.45 36 2.827833e+10 125144.40
## 8390) population>=982.5 23 1.461229e+10 117382.60
## 16780) sdbedrooms< 1.097202 11 6.456867e+09 106354.50 *
## 16781) sdbedrooms>=1.097202 12 5.591309e+09 127491.70 *
## 8391) population< 982.5 13 9.828823e+09 138876.90 *
## 1049) sdbedrooms>=1.359339 26 4.998054e+10 128642.30
## 2098) sdrooms>=5.743115 17 1.257372e+10 113888.20 *
## 2099) sdrooms< 5.743115 9 2.671619e+10 156511.10 *
## 525) sdbedrooms< 0.9317544 8 1.228635e+11 162875.00 *
## 263) longitude< -121.73 203 5.423242e+11 119556.20
## 526) latitude>=37.735 178 2.914599e+11 109251.70
## 1052) medianIncome< 1.89275 78 3.777675e+10 89521.79
## 2104) longitude< -122.295 40 8.258850e+09 78602.50 *
## 2105) longitude>=-122.295 38 1.972841e+10 101015.80
## 4210) sdrooms>=4.540244 30 7.426634e+09 95356.67 *
## 4211) sdrooms< 4.540244 8 7.738119e+09 122237.50 *
## 1053) medianIncome>=1.89275 100 1.996371e+11 124641.00
## 2106) longitude>=-122.225 31 1.605805e+10 105180.60
## 4212) medianIncome< 2.11015 13 5.663031e+08 94476.92 *
## 4213) medianIncome>=2.11015 18 1.292666e+10 112911.10 *
## 2107) longitude< -122.225 69 1.665648e+11 133384.10
## 4214) medianIncome>=2.001 55 8.296422e+10 124700.00
## 8428) medianIncome< 2.36255 46 5.789684e+10 117215.20
## 16856) latitude>=38.65 12 1.810077e+10 99866.67 *
## 16857) latitude< 38.65 34 3.490970e+10 123338.20
## 33714) population>=1131.5 10 1.230201e+09 105330.00 *
## 33715) population< 1131.5 24 2.908530e+10 130841.70
## 67430) latitude< 38.275 17 1.988390e+10 123182.40 *
## 67431) latitude>=38.275 7 5.782057e+09 149442.90 *
## 8429) medianIncome>=2.36255 9 9.318982e+09 162955.60 *
## 4215) medianIncome< 2.001 14 6.315828e+10 167500.00 *
## 527) latitude< 37.735 25 9.739281e+10 192924.00
## 1054) housingMedianAge< 19 7 1.421007e+10 141685.70 *
## 1055) housingMedianAge>=19 18 5.765838e+10 212850.00 *
## 33) latitude< 35.305 530 2.395649e+12 115509.60
## 66) longitude>=-117.355 273 4.756575e+11 93961.54
## 132) longitude>=-116.525 73 7.244954e+10 69542.47
## 264) sdbedrooms>=1.026908 66 2.326990e+10 65539.39
## 528) longitude>=-116.34 53 1.279774e+10 61800.00
## 1056) medianIncome< 1.44605 10 1.407356e+09 48880.00 *
## 1057) medianIncome>=1.44605 43 9.332919e+09 64804.65
## 2114) latitude>=32.895 35 4.845886e+09 61042.86 *
## 2115) latitude< 32.895 8 1.824859e+09 81262.50 *
## 529) longitude< -116.34 13 6.709637e+09 80784.62 *
## 265) sdbedrooms< 1.026908 7 3.815015e+10 107285.70 *
## 133) longitude< -116.525 200 3.437905e+11 102874.50
## 266) latitude>=33.685 130 8.870547e+10 88033.08
## 532) sdrooms< 6.768876 120 5.059127e+10 85534.17
## 1064) housingMedianAge>=20.5 74 1.707397e+10 78398.65
## 2128) population< 2608.5 67 1.289061e+10 76673.13
## 4256) sdrooms< 5.267666 49 6.971640e+09 73414.29 *
## 4257) sdrooms>=5.267666 18 3.981984e+09 85544.44 *
## 2129) population>=2608.5 7 2.074509e+09 94914.29 *
## 1065) housingMedianAge< 20.5 46 2.368837e+10 97013.04
## 2130) longitude>=-116.905 8 5.253935e+09 79325.00 *
## 2131) longitude< -116.905 38 1.540457e+10 100736.80
## 4262) sdrooms< 5.144522 19 9.213164e+09 92163.16 *
## 4263) sdrooms>=5.144522 19 3.398098e+09 109310.50 *
## 533) sdrooms>=6.768876 10 2.837270e+10 118020.00 *
## 267) latitude< 33.685 70 1.732713e+11 130437.10
## 534) latitude< 32.715 24 1.015334e+10 97958.33
## 1068) medianIncome< 1.826 10 7.094800e+08 81900.00 *
## 1069) medianIncome>=1.826 14 5.023229e+09 109428.60 *
## 535) latitude>=32.715 46 1.245921e+11 147382.60
## 1070) population>=544.5 39 5.217812e+10 135843.60
## 2140) housingMedianAge< 35.5 31 2.390293e+10 128422.60
## 4280) latitude< 32.755 9 2.280702e+09 108044.40 *
## 4281) latitude>=32.755 22 1.635587e+10 136759.10
## 8562) population>=1062 13 7.517991e+09 127038.50 *
## 8563) population< 1062 9 5.835180e+09 150800.00 *
## 2141) housingMedianAge>=35.5 8 1.995252e+10 164600.00 *
## 1071) population< 544.5 7 3.828983e+10 211671.40 *
## 67) longitude< -117.355 257 1.658581e+12 138399.20
## 134) latitude>=33.605 249 1.017024e+12 131671.50
## 268) population>=237.5 241 5.491026e+11 124821.60
## 536) longitude>=-118.315 181 1.826615e+11 113814.40
## 1072) latitude>=33.875 165 1.042311e+11 110154.50
## 2144) medianIncome< 2.1669 116 4.228545e+10 104587.10
## 4288) latitude>=34.38 9 3.315160e+09 75600.00 *
## 4289) latitude< 34.38 107 3.077196e+10 107025.20
## 8578) longitude< -118.205 80 1.043382e+10 103477.50
## 17156) longitude>=-118.285 50 2.141210e+09 97902.00 *
## 17157) longitude< -118.285 30 4.147783e+09 112770.00 *
## 8579) longitude>=-118.205 27 1.634778e+10 117537.00
## 17158) longitude>=-117.74 14 5.156969e+09 103071.40 *
## 17159) longitude< -117.74 13 5.106357e+09 133115.40 *
## 2145) medianIncome>=2.1669 49 4.983791e+10 123334.70
## 4290) longitude>=-117.7 17 9.595865e+09 103217.60 *
## 4291) longitude< -117.7 32 2.970731e+10 134021.90
## 8582) longitude< -118.205 12 6.264402e+09 112175.00 *
## 8583) longitude>=-118.205 20 1.427902e+10 147130.00
## 17166) sdrooms>=4.47115 13 3.749068e+09 134769.20 *
## 17167) sdrooms< 4.47115 7 4.854949e+09 170085.70 *
## 1073) latitude< 33.875 16 5.342918e+10 151556.20 *
## 537) longitude< -118.315 60 2.783567e+11 158026.70
## 1074) population>=529.5 51 1.793040e+11 147394.10
## 2148) longitude>=-120.525 43 1.439043e+11 138323.30
## 4296) latitude>=35.045 8 5.193089e+09 61087.50 *
## 4297) latitude< 35.045 35 8.008022e+10 155977.10
## 8594) sdrooms>=4.667819 19 1.384282e+10 135431.60 *
## 8595) sdrooms< 4.667819 16 4.869303e+10 180375.00 *
## 2149) longitude< -120.525 8 1.284458e+10 196150.00 *
## 1075) population< 529.5 9 6.061532e+10 218277.80 *
## 269) population< 237.5 8 1.159583e+11 338025.10 *
## 135) latitude< 33.605 8 2.794966e+11 347800.40 *
## 17) medianIncome>=2.4187 1748 7.789670e+12 138852.60
## 34) latitude>=34.455 1100 3.919012e+12 125499.50
## 68) longitude>=-121.695 696 1.189826e+12 107261.10
## 136) longitude>=-119.915 225 1.303636e+11 83179.11
## 272) latitude< 37.23 211 1.064963e+11 80825.59
## 544) latitude>=34.845 192 7.687962e+10 77998.96
## 1088) sdrooms< 5.654617 138 2.705156e+10 74917.39
## 2176) housingMedianAge>=27.5 68 1.025123e+10 69648.53 *
## 2177) housingMedianAge< 27.5 70 1.307878e+10 80035.71 *
## 1089) sdrooms>=5.654617 54 4.516866e+10 85874.07
## 2178) longitude>=-118.945 11 3.937605e+09 64263.64 *
## 2179) longitude< -118.945 43 3.477979e+10 91402.33
## 4358) longitude< -119.685 11 3.205367e+09 75954.55 *
## 4359) longitude>=-119.685 32 2.804712e+10 96712.50
## 8718) population>=659.5 24 1.154184e+10 90200.00
## 17436) population< 1264 12 1.586067e+09 77433.33 *
## 17437) population>=1264 12 6.044067e+09 102966.70 *
## 8719) population< 659.5 8 1.243366e+10 116250.00 *
## 545) latitude< 34.845 19 1.258064e+10 109389.50 *
## 273) latitude>=37.23 14 5.083995e+09 118650.00 *
## 137) longitude< -119.915 471 8.666410e+11 118765.20
## 274) latitude>=35.845 432 5.630769e+11 113542.40
## 548) latitude>=39.315 41 2.630355e+10 87385.37
## 1096) population< 1088.5 32 1.336024e+10 80206.25
## 2192) longitude>=-121.555 23 6.866612e+09 73734.78
## 4384) medianIncome< 2.76565 16 2.828498e+09 67262.50 *
## 4385) medianIncome>=2.76565 7 1.835874e+09 88528.57 *
## 2193) longitude< -121.555 9 3.068782e+09 96744.44 *
## 1097) population>=1088.5 9 5.429969e+09 112911.10 *
## 549) latitude< 39.315 391 5.057802e+11 116285.20
## 1098) sdrooms< 6.027356 311 3.778234e+11 111653.10
## 2196) longitude>=-120.8 59 3.731302e+10 93003.39
## 4392) latitude< 37.815 37 1.501292e+10 80318.92
## 8784) medianIncome< 2.9758 27 5.454341e+09 75618.52 *
## 8785) medianIncome>=2.9758 10 7.351409e+09 93010.00 *
## 4393) latitude>=37.815 22 6.334851e+09 114336.40
## 8786) housingMedianAge>=20 10 1.819896e+09 102920.00 *
## 8787) housingMedianAge< 20 12 2.125510e+09 123850.00 *
## 2197) longitude< -120.8 252 3.151851e+11 116019.40
## 4394) latitude>=37.555 222 2.244559e+11 112232.40
## 8788) housingMedianAge>=34.5 78 4.575086e+10 96671.79
## 17576) latitude< 38.545 41 1.484213e+10 87365.85
## 35152) latitude>=37.86 28 9.095649e+09 81042.86 *
## 35153) latitude< 37.86 13 2.215917e+09 100984.60 *
## 17577) latitude>=38.545 37 2.342363e+10 106983.80
## 35154) latitude>=38.615 19 9.649557e+09 93973.68 *
## 35155) latitude< 38.615 18 7.163425e+09 120716.70 *
## 8789) housingMedianAge< 34.5 144 1.495885e+11 120661.10
## 17578) longitude< -121.275 80 6.795896e+10 113115.00
## 35156) latitude>=38.715 12 3.215929e+09 88041.67 *
## 35157) latitude< 38.715 68 5.586766e+10 117539.70
## 70314) latitude< 38.545 31 1.854876e+10 104658.10
## 140628) population>=698.5 24 9.190565e+09 98125.00 *
## 140629) population< 698.5 7 4.821817e+09 127057.10 *
## 70315) latitude>=38.545 37 2.786500e+10 128332.40
## 140630) longitude>=-121.315 9 2.246080e+09 112166.70 *
## 140631) longitude< -121.315 28 2.251094e+10 133528.60
## 281262) population>=1100.5 21 1.301925e+10 127785.70
## 562524) longitude< -121.345 10 1.765761e+09 113470.00 *
## 562525) longitude>=-121.345 11 7.341000e+09 140800.00 *
## 281263) population< 1100.5 7 6.721337e+09 150757.10 *
## 17579) longitude>=-121.275 64 7.137962e+10 130093.80
## 35158) sdrooms< 5.114486 26 1.597129e+10 114703.80
## 70316) longitude>=-121.005 12 4.562889e+09 103941.70 *
## 70317) longitude< -121.005 14 8.827169e+09 123928.60 *
## 35159) sdrooms>=5.114486 38 4.503683e+10 140623.70
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## 12883) sdbedrooms>=1.034596 14 2.305805e+10 266007.10 *
## 3221) longitude< -119.715 94 2.396354e+11 273963.80
## 6442) latitude>=36.68 84 1.615395e+11 268035.70
## 12884) population< 2790 72 1.142157e+11 262436.10
## 25768) longitude>=-121.875 49 8.014024e+10 253255.10
## 51536) sdbedrooms< 1.053076 42 5.359412e+10 246219.00
## 103072) longitude>=-121.75 7 2.926518e+10 217642.90 *
## 103073) longitude< -121.75 35 1.746952e+10 251934.30
## 206146) sdrooms< 5.906565 11 6.388849e+09 232909.10 *
## 206147) sdrooms>=5.906565 24 5.274260e+09 260654.20 *
## 51537) sdbedrooms>=1.053076 7 1.199133e+10 295471.40 *
## 25769) longitude< -121.875 23 2.114593e+10 281995.70
## 51538) longitude< -121.905 15 1.152056e+10 270013.30 *
## 51539) longitude>=-121.905 8 3.433659e+09 304462.50 *
## 12885) population>=2790 12 3.152059e+10 301633.30 *
## 6443) latitude< 36.68 10 5.034740e+10 323760.00 *
## 1611) latitude< 34.255 166 4.573332e+11 288659.70
## 3222) longitude>=-118.045 108 2.027302e+11 278784.30
## 6444) latitude>=33.665 97 1.373776e+11 274596.90
## 12888) housingMedianAge>=25.5 23 1.559147e+10 251587.00
## 25776) medianIncome< 6.11745 14 4.449477e+09 237814.30 *
## 25777) medianIncome>=6.11745 9 4.355429e+09 273011.10 *
## 12889) housingMedianAge< 25.5 74 1.058237e+11 281748.60
## 25778) latitude< 33.895 52 4.015165e+10 273451.90
## 51556) sdrooms< 6.951643 44 3.018335e+10 268586.40
## 103112) sdbedrooms>=1.016861 28 1.469936e+10 261628.60
## 206224) housingMedianAge>=18.5 16 7.097744e+09 253181.20 *
## 206225) housingMedianAge< 18.5 12 4.937609e+09 272891.70 *
## 103113) sdbedrooms< 1.016861 16 1.175636e+10 280762.50 *
## 51557) sdrooms>=6.951643 8 3.197629e+09 300212.50 *
## 25779) latitude>=33.895 22 5.363205e+10 301359.10
## 51558) longitude>=-117.875 9 2.445922e+09 272355.60 *
## 51559) longitude< -117.875 13 3.837393e+10 321438.50 *
## 6445) latitude< 33.665 11 4.865391e+10 315709.10 *
## 3223) longitude< -118.045 58 2.244583e+11 307048.30
## 6446) sdrooms>=5.795841 40 8.660898e+10 292317.50
## 12892) longitude< -118.075 32 5.165816e+10 281284.40
## 25784) medianIncome< 6.1174 14 1.875162e+10 257250.00 *
## 25785) medianIncome>=6.1174 18 1.852945e+10 299977.80 *
## 12893) longitude>=-118.075 8 1.547404e+10 336450.00 *
## 6447) sdrooms< 5.795841 18 1.098808e+11 339783.40 *
## 403) longitude< -121.955 41 2.089266e+11 332914.70
## 806) medianIncome< 5.88715 14 2.316290e+10 283778.60 *
## 807) medianIncome>=5.88715 27 1.344363e+11 358392.70
## 1614) housingMedianAge< 16.5 7 1.745739e+10 299085.70 *
## 1615) housingMedianAge>=16.5 20 8.374025e+10 379150.10
## 3230) latitude>=37.615 8 7.854879e+09 342662.50 *
## 3231) latitude< 37.615 12 5.813411e+10 403475.20 *
## 101) sdbedrooms>=1.094828 92 9.735530e+11 320784.90
## 202) sdrooms>=8.209351 14 3.701241e+10 207942.90 *
## 203) sdrooms< 8.209351 78 7.262774e+11 341038.60
## 406) housingMedianAge< 7.5 23 1.067022e+11 287517.40
## 812) sdrooms>=6.552914 12 1.691733e+10 242191.70 *
## 813) sdrooms< 6.552914 11 3.823743e+10 336963.70 *
## 407) housingMedianAge>=7.5 55 5.261399e+11 363420.20
## 814) sdbedrooms< 1.127171 22 2.248234e+11 316372.80
## 1628) housingMedianAge>=16.5 15 1.039404e+11 280826.70 *
## 1629) housingMedianAge< 16.5 7 6.131687e+10 392543.00 *
## 815) sdbedrooms>=1.127171 33 2.201565e+11 394785.10
## 1630) population< 1450.5 25 1.208366e+11 365240.10
## 3260) sdbedrooms< 1.17798 10 2.992650e+10 316050.00 *
## 3261) sdbedrooms>=1.17798 15 5.058236e+10 398033.50 *
## 1631) population>=1450.5 8 9.301089e+09 487113.40 *
## 51) housingMedianAge>=27.5 210 1.678177e+12 334980.60
## 102) sdbedrooms< 1.046954 151 1.034068e+12 316811.30
## 204) longitude>=-122.105 121 7.108451e+11 299738.10
## 408) longitude>=-118 22 1.004546e+11 253227.30
## 816) sdbedrooms< 1.00369 13 2.806891e+10 236961.50 *
## 817) sdbedrooms>=1.00369 9 6.397806e+10 276722.30 *
## 409) longitude< -118 99 5.522232e+11 310073.80
## 818) latitude>=37.375 13 2.767636e+10 240400.00 *
## 819) latitude< 37.375 86 4.518996e+11 320605.90
## 1638) sdrooms< 6.699118 79 3.751150e+11 314683.60
## 3276) longitude>=-118.355 25 8.062294e+10 288192.00
## 6552) sdrooms< 6.20891 16 1.647896e+10 263262.50 *
## 6553) sdrooms>=6.20891 9 3.652265e+10 332511.10 *
## 3277) longitude< -118.355 54 2.688243e+11 326948.20
## 6554) latitude>=34.165 43 1.377238e+11 308072.10
## 13108) latitude< 34.34 14 3.577636e+10 264100.00 *
## 13109) latitude>=34.34 29 6.180972e+10 329300.00
## 26218) sdbedrooms>=1.019097 7 1.560331e+10 296814.30 *
## 26219) sdbedrooms< 1.019097 22 3.646867e+10 339636.40
## 52438) latitude>=37.285 14 1.431720e+10 323285.70 *
## 52439) latitude< 37.285 8 1.185874e+10 368250.00 *
## 6555) latitude< 34.165 11 5.588732e+10 400736.50 *
## 1639) sdrooms>=6.699118 7 4.274327e+10 387443.10 *
## 205) longitude< -122.105 30 1.456917e+11 385673.40
## 410) medianIncome< 6.0829 14 4.645001e+10 339471.40 *
## 411) medianIncome>=6.0829 16 4.320778e+10 426100.20 *
## 103) sdbedrooms>=1.046954 59 4.666827e+11 381481.50
## 206) longitude>=-118.375 21 1.472166e+11 343657.30
## 412) population< 864.5 14 5.263004e+10 315457.20 *
## 413) population>=864.5 7 6.118630e+10 400057.40 *
## 207) longitude< -118.375 38 2.728186e+11 402384.40
## 414) latitude>=34.145 25 1.693173e+11 371200.20
## 828) sdbedrooms< 1.117885 18 9.260783e+10 354266.70 *
## 829) sdbedrooms>=1.117885 7 5.827609e+10 414743.30 *
## 415) latitude< 34.145 13 3.243707e+10 462354.20 *
## 13) housingMedianAge>=36.5 322 3.164933e+12 350979.30
## 26) medianIncome< 5.7032 186 1.636538e+12 324271.60
## 52) longitude>=-122.295 153 1.252022e+12 311822.30
## 104) sdbedrooms< 0.9277427 8 4.142589e+10 223887.50 *
## 105) sdbedrooms>=0.9277427 145 1.145323e+12 316673.80
## 210) latitude>=33.78 137 1.016786e+12 312162.10
## 420) longitude>=-118.365 62 3.877134e+11 286006.50
## 840) latitude< 34.105 30 1.004423e+11 234366.70
## 1680) housingMedianAge< 45.5 20 3.811017e+10 218615.00
## 3360) medianIncome>=5.2151 10 8.882361e+09 198230.00 *
## 3361) medianIncome< 5.2151 10 2.091684e+10 239000.00 *
## 1681) housingMedianAge>=45.5 10 4.744524e+10 265870.00 *
## 841) latitude>=34.105 32 1.322710e+11 334418.80
## 1682) latitude>=34.185 7 1.834073e+10 278685.70 *
## 1683) latitude< 34.185 25 8.609901e+10 350024.00
## 3366) medianIncome>=5.41725 10 1.151906e+10 313750.00 *
## 3367) medianIncome< 5.41725 15 5.264989e+10 374206.70 *
## 421) longitude< -118.365 75 5.515938e+11 333784.10
## 842) latitude>=34.185 49 2.189705e+11 295353.10
## 1684) latitude>=37.88 11 2.603157e+10 251790.90 *
## 1685) latitude< 37.88 38 1.660221e+11 307963.20
## 3370) population< 983.5 31 1.249733e+11 296951.60
## 6740) medianIncome< 5.41485 24 9.770217e+10 284787.50
## 13480) longitude>=-122.215 17 4.821489e+10 266594.10 *
## 13481) longitude< -122.215 7 3.019475e+10 328971.40 *
## 6741) medianIncome>=5.41485 7 1.154448e+10 338657.10 *
## 3371) population>=983.5 7 2.064349e+10 356728.60 *
## 843) latitude< 34.185 26 1.238632e+11 406211.70
## 1686) sdbedrooms< 0.9980695 8 2.227606e+10 353450.00 *
## 1687) sdbedrooms>=0.9980695 18 6.941875e+10 429661.40 *
## 211) latitude< 33.78 8 7.799064e+10 393937.90 *
## 53) longitude< -122.295 33 2.508620e+11 381991.10
## 106) housingMedianAge< 46.5 14 6.429031e+10 328207.10 *
## 107) housingMedianAge>=46.5 19 1.162330e+11 421621.40 *
## 27) medianIncome>=5.7032 136 1.214269e+12 387506.10
## 54) population< 292 9 1.955410e+11 269555.80 *
## 55) population>=292 127 8.846443e+11 395864.70
## 110) latitude>=34.155 66 4.902087e+11 368059.20
## 220) longitude>=-122.39 49 2.829100e+11 339857.20
## 440) longitude>=-121.27 15 1.323146e+11 307786.70 *
## 441) longitude< -121.27 34 1.283613e+11 354005.90
## 882) sdbedrooms< 1.062357 24 6.958878e+10 335954.20
## 1764) population< 828.5 9 2.561960e+10 305644.40 *
## 1765) population>=828.5 15 3.074020e+10 354140.00 *
## 883) sdbedrooms>=1.062357 10 3.218193e+10 397330.10 *
## 221) longitude< -122.39 17 5.599391e+10 449347.50 *
## 111) latitude< 34.155 61 2.881977e+11 425949.40
## 222) longitude>=-118.415 46 2.310759e+11 414067.60
## 444) housingMedianAge< 48.5 31 1.682165e+11 396274.30
## 888) longitude>=-118.375 14 7.252548e+10 364492.90 *
## 889) longitude< -118.375 17 6.990491e+10 422447.20 *
## 445) housingMedianAge>=48.5 15 3.276129e+10 450840.30 *
## 223) longitude< -118.415 15 3.071213e+10 462387.00 *
## 7) medianIncome>=6.54555 1122 9.530861e+12 409761.20
## 14) medianIncome< 7.81515 569 4.554581e+12 364621.10
## 28) housingMedianAge< 26.5 356 2.148649e+12 333757.70
## 56) sdbedrooms< 1.030796 207 8.265221e+11 317365.20
## 112) longitude>=-122.035 189 6.496536e+11 311699.50
## 224) sdbedrooms< 0.9559794 35 1.140015e+11 271560.00
## 448) longitude< -118.73 15 5.752287e+10 243473.30 *
## 449) longitude>=-118.73 20 3.577104e+10 292625.00
## 898) longitude>=-117.735 11 2.341444e+10 275900.00 *
## 899) longitude< -117.735 9 5.518860e+09 313066.70 *
## 225) sdbedrooms>=0.9559794 154 4.664446e+11 320822.10
## 450) longitude>=-117.89 51 1.398023e+11 295151.00
## 900) medianIncome< 6.98355 29 3.242942e+10 277372.40
## 1800) longitude>=-117.35 7 5.922437e+09 254057.10 *
## 1801) longitude< -117.35 22 2.149102e+10 284790.90
## 3602) housingMedianAge>=14.5 15 8.616964e+09 272620.00 *
## 3603) housingMedianAge< 14.5 7 5.890734e+09 310871.40 *
## 901) medianIncome>=6.98355 22 8.612385e+10 318586.40
## 1802) latitude>=33.59 15 3.639001e+10 296133.30 *
## 1803) latitude< 33.59 7 2.596721e+10 366700.10 *
## 451) longitude< -117.89 103 2.763915e+11 333533.00
## 902) latitude>=37.365 29 5.308426e+10 302327.60
## 1804) population< 1020 11 1.153853e+10 271245.50 *
## 1805) population>=1020 18 2.442431e+10 321322.20 *
## 903) latitude< 37.365 74 1.840008e+11 345762.20
## 1806) medianIncome< 7.0551 40 1.167298e+11 332197.50
## 3612) longitude< -118.545 18 3.653066e+10 311672.20 *
## 3613) longitude>=-118.545 22 6.641152e+10 348991.00
## 7226) population< 2187.5 14 2.481284e+10 330214.30 *
## 7227) population>=2187.5 8 2.802500e+10 381850.10 *
## 1807) medianIncome>=7.0551 34 5.125217e+10 361720.60
## 3614) population>=1715 14 1.343076e+10 346750.00 *
## 3615) population< 1715 20 3.248738e+10 372200.00
## 7230) medianIncome< 7.39915 11 8.589136e+09 360618.20 *
## 7231) medianIncome>=7.39915 9 2.061927e+10 386355.70 *
## 113) longitude< -122.035 18 1.070975e+11 376855.70 *
## 57) sdbedrooms>=1.030796 149 1.189228e+12 356531.10
## 114) housingMedianAge< 18.5 105 6.893702e+11 337038.20
## 228) sdbedrooms< 1.06133 36 1.648442e+11 308819.50
## 456) sdbedrooms>=1.038086 29 9.670475e+10 292644.80
## 912) housingMedianAge>=10 17 6.035240e+10 270329.40 *
## 913) housingMedianAge< 10 12 1.589377e+10 324258.30 *
## 457) sdbedrooms< 1.038086 7 2.912086e+10 375828.70 *
## 229) sdbedrooms>=1.06133 69 4.809027e+11 351761.00
## 458) medianIncome< 7.453 54 3.349777e+11 339568.60
## 916) sdrooms>=9.276739 7 2.955466e+10 271742.90 *
## 917) sdrooms< 9.276739 47 2.684246e+11 349670.30
## 1834) sdrooms< 8.560451 40 1.793038e+11 338832.60
## 3668) latitude< 33.925 19 5.451410e+10 309889.50 *
## 3669) latitude>=33.925 21 9.447286e+10 365019.20
## 7338) latitude>=37.195 13 2.285246e+10 333615.40 *
## 7339) latitude< 37.195 8 3.796636e+10 416050.40 *
## 1835) sdrooms>=8.560451 7 5.757517e+10 411600.40 *
## 459) medianIncome>=7.453 15 1.089992e+11 395653.70 *
## 115) housingMedianAge>=18.5 44 3.647521e+11 403048.10
## 230) sdrooms>=7.46821 17 9.282697e+10 354253.00 *
## 231) sdrooms< 7.46821 27 2.059638e+11 433770.90
## 462) population< 668.5 7 9.328162e+10 372571.90 *
## 463) population>=668.5 20 7.728886e+10 455190.60
## 926) housingMedianAge< 24.5 13 6.006220e+10 436277.30 *
## 927) housingMedianAge>=24.5 7 3.940265e+09 490315.10 *
## 29) housingMedianAge>=26.5 213 1.500052e+12 416205.00
## 58) latitude>=37.61 40 2.145443e+11 365105.10
## 116) longitude>=-122.41 31 1.314318e+11 345451.60
## 232) sdrooms< 6.742063 8 2.327935e+10 280212.50 *
## 233) sdrooms>=6.742063 23 6.226018e+10 368143.50
## 466) population>=767 15 3.647306e+10 354013.30 *
## 467) population< 767 8 1.717674e+10 394637.50 *
## 117) longitude< -122.41 9 2.989458e+10 432800.30 *
## 59) latitude< 37.61 173 1.156910e+12 428020.00
## 118) longitude>=-117.91 24 2.145605e+11 356721.00
## 236) latitude>=33.67 12 3.754812e+10 301775.00 *
## 237) latitude< 33.67 12 1.045549e+11 411667.00 *
## 119) longitude< -117.91 149 8.006922e+11 439504.40
## 238) longitude>=-122.055 118 7.096350e+11 428265.60
## 476) latitude>=34.145 50 2.288876e+11 403498.10
## 952) sdrooms< 6.580794 18 6.088328e+10 348883.30 *
## 953) sdrooms>=6.580794 32 8.411373e+10 434219.00
## 1906) medianIncome< 6.85985 14 3.774903e+10 402814.40 *
## 1907) medianIncome>=6.85985 18 2.181816e+10 458644.70 *
## 477) latitude< 34.145 68 4.275235e+11 446477.00
## 954) longitude>=-118.405 51 3.625876e+11 428635.60
## 1908) sdrooms< 5.716681 8 1.166315e+11 358237.60 *
## 1909) sdrooms>=5.716681 43 1.989328e+11 441733.00
## 3818) longitude>=-118.145 18 1.018126e+11 414255.80 *
## 3819) longitude< -118.145 25 7.374562e+10 461516.50
## 7638) latitude< 34.02 16 5.561997e+10 441800.20 *
## 7639) latitude>=34.02 9 8.487749e+08 496567.60 *
## 955) longitude< -118.405 17 0.000000e+00 500001.00 *
## 239) longitude< -122.055 31 1.941842e+10 482284.50
## 478) population>=1352 7 2.810477e+09 458771.60 *
## 479) population< 1352 24 1.160919e+10 489142.40
## 958) longitude< -122.155 10 8.717974e+09 477910.60 *
## 959) longitude>=-122.155 14 7.285802e+08 497165.10 *
## 15) medianIncome>=7.81515 553 2.623915e+12 456207.30
## 30) medianIncome< 9.0788 271 1.501310e+12 430448.70
## 60) housingMedianAge< 29.5 159 8.888456e+11 399836.10
## 120) longitude>=-118.07 66 2.373473e+11 373188.00
## 240) sdrooms< 8.927196 59 1.692554e+11 364188.20
## 480) population< 918 15 7.120167e+10 331786.70 *
## 481) population>=918 44 7.693733e+10 375234.10
## 962) medianIncome< 8.77915 36 4.460083e+10 368352.80
## 1924) latitude>=33.56 28 2.635977e+10 361310.70
## 3848) latitude< 33.705 11 5.907462e+09 337827.30 *
## 3849) latitude>=33.705 17 1.046093e+10 376505.90 *
## 1925) latitude< 33.56 8 1.199264e+10 393000.00 *
## 963) medianIncome>=8.77915 8 2.296065e+10 406200.10 *
## 241) sdrooms>=8.927196 7 2.303496e+10 449043.30 *
## 121) longitude< -118.07 93 5.713691e+11 418747.60
## 242) latitude>=34.135 78 4.571694e+11 403413.00
## 484) longitude>=-121.485 33 2.658778e+11 370782.00
## 968) latitude>=34.895 10 5.984532e+10 284940.00 *
## 969) latitude< 34.895 23 1.003056e+11 408104.60
## 1938) medianIncome< 8.5471 15 3.697653e+10 372666.70 *
## 1939) medianIncome>=8.5471 8 9.170775e+09 474550.60 *
## 485) longitude< -121.485 45 1.303859e+11 427342.40
## 970) latitude>=37.815 15 3.043296e+10 389580.00 *
## 971) latitude< 37.815 30 6.786792e+10 446223.60
## 1942) population>=1298 14 1.946593e+10 420428.60 *
## 1943) population< 1298 16 3.093573e+10 468794.20 *
## 243) latitude< 34.135 15 4.809797e+08 498487.60 *
## 61) housingMedianAge>=29.5 112 2.519260e+11 473907.80
## 122) latitude>=37.805 22 5.695861e+10 434050.20
## 244) sdbedrooms< 1.031657 13 2.358685e+10 409792.30 *
## 245) sdbedrooms>=1.031657 9 1.467224e+10 469089.40 *
## 123) latitude< 37.805 90 1.514743e+11 483650.80
## 246) sdbedrooms< 1.032476 49 1.309183e+11 472812.90
## 492) medianIncome< 8.10515 21 9.909855e+10 450352.90
## 984) housingMedianAge< 35.5 8 4.075518e+10 405212.60 *
## 985) housingMedianAge>=35.5 13 3.201078e+10 478131.50 *
## 493) medianIncome>=8.10515 28 1.328110e+10 489657.90
## 986) sdbedrooms< 0.9659991 7 6.256877e+09 468614.70 *
## 987) sdbedrooms>=0.9659991 21 2.891269e+09 496672.30 *
## 247) sdbedrooms>=1.032476 41 7.921961e+09 496603.30 *
## 31) medianIncome>=9.0788 282 7.700007e+11 480961.10
## 62) population< 56.5 7 1.188895e+11 303543.00 *
## 63) population>=56.5 275 4.251624e+11 485477.20
## 126) housingMedianAge< 18.5 73 2.422440e+11 464059.50
## 252) medianIncome< 11.06295 50 2.043246e+11 452074.50
## 504) sdbedrooms< 0.9643377 7 4.738186e+10 388343.00 *
## 505) sdbedrooms>=0.9643377 43 1.238824e+11 462449.30
## 1010) longitude>=-121.82 32 1.048125e+11 451881.60
## 2020) population< 1479 18 8.112218e+10 434428.10 *
## 2021) population>=1479 14 1.115713e+10 474321.90 *
## 1011) longitude< -121.82 11 5.100145e+09 493191.80 *
## 253) medianIncome>=11.06295 23 1.512434e+10 490113.80
## 506) population< 773 7 1.100955e+10 470372.00 *
## 507) population>=773 16 1.930350e+08 498750.90 *
## 127) housingMedianAge>=18.5 202 1.373304e+11 493217.20
## 254) sdrooms< 6.286722 12 5.995560e+10 453675.70 *
## 255) sdrooms>=6.286722 190 5.742733e+10 495714.60
## 510) medianIncome< 10.1328 54 4.424725e+10 486774.90
## 1020) longitude>=-117.93 9 1.273603e+10 456011.40 *
## 1021) longitude< -117.93 45 2.129024e+10 492927.50
## 2042) population< 884 12 1.620078e+10 478359.00 *
## 2043) population>=884 33 1.616411e+09 498225.20 *
## 511) medianIncome>=10.1328 136 7.150874e+09 499264.20 *##
## Call:
## randomForest(formula = medianHouseValue ~ longitude + latitude + housingMedianAge + sdrooms + sdbedrooms + population + medianIncome, data = cahousing_train, importance = TRUE)
## Type of random forest: regression
## Number of trees: 500
## No. of variables tried at each split: 2
##
## Mean of squared residuals: 2709298892
## % Var explained: 79.54## Distribution not specified, assuming gaussian ...## gbm(formula = medianHouseValue ~ longitude + latitude + housingMedianAge +
## sdrooms + sdbedrooms + population + medianIncome, data = cahousing_train)
## A gradient boosted model with gaussian loss function.
## 100 iterations were performed.
## There were 7 predictors of which 6 had non-zero influence.Now, we estimate their RMSE to find out which model is our best predictive model.
## [1] 72985.15## [1] 70109.68## [1] 59942.64## [1] 51163.03## [1] 74194.08According to our results, we can find the Random Forecast model has lower RMSE, so we use this to build our best predictive model.
We can find that above graphs are perform well, because Figure1: Actual Median House Value in California are almost match with Figure 4-2: Prediction Value in California. Therefore, we can indicate the random forest is the best predictive model in this case. In conclusion, our predictive model is effective, then we may use this predictive model to forecast other states or area in United State.